You do NOT take this type of faucet out/ or apart from the underside. Remove the handle and trim, and there will be a nut holding it in. Take that off and the valve drops DOWN. A photo of the top would help us explain

Tried this, but my basin wrench wouldn't get wide enough to grip the "nodes".
I'm having a little success removing them from the top, as jimbo advised, but the whole section spins.
As you can see, the faucets are corroded and quite old. I'm having a really hard time getting any twist or progress.
I've been using vice grips on the geared protrusion shown to hold the unit in place while I use a crescent wrench, hair dryer, wd40 (all I've got at home), and some hammering. Painfully slow.

By the way, that nut is an electrical lock nut, which should be regular pipe thread. A pair of pliers should be able to grab it, but its difficult to figure out from here if there is enough room to fit in that area. You will have to be careful not to damage the basin, but you MAY be able to jam something like a large screwdriver in there to keep the nut from turning and try turning the valve body again, but as Jimb said, it may not be removable from the top up, as the valve body casting and it port may not fit through the hole. A Fein oscillating saw or cheap knockoff will be able to cut that inlet port off and allow it to fit through the hole Safety Googles are a must here.

That is NOT an electrical locknut, nor is it a "pipe thread". It is a "proprietary" locknut designed by the faucet manufacturer for THEIR faucet.

Click to expand...

To a hammer, everything looks like a nail.

Still, it looks like some cutting will be involved. On the other hand, if this sink basin is not very expensive, it may just be better to start with a new sink and new faucet. it's easy for me to spend your money, lol